Optimal. Leaf size=437 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}-\frac{8 a f-x \left (a g+11 b c+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{x \left (7 (a g+11 b c)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (-a g+b c+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.531377, antiderivative size = 437, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {1858, 1854, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}-\frac{8 a f-x \left (a g+11 b c+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{x \left (7 (a g+11 b c)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (-a g+b c+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1858
Rule 1854
Rule 1855
Rule 1876
Rule 275
Rule 205
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^4} \, dx &=\frac{x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}-\frac{\int \frac{-11 b c-a g-10 b d x-9 b e x^2-8 b f x^3}{\left (a+b x^4\right )^3} \, dx}{12 a b}\\ &=\frac{x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}-\frac{8 a f-x \left (11 b c+a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{\int \frac{-7 (-11 b c-a g)+60 b d x+45 b e x^2}{\left (a+b x^4\right )^2} \, dx}{96 a^2 b}\\ &=\frac{x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac{\int \frac{-21 (11 b c+a g)-120 b d x-45 b e x^2}{a+b x^4} \, dx}{384 a^3 b}\\ &=\frac{x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac{\int \left (-\frac{120 b d x}{a+b x^4}+\frac{-21 (11 b c+a g)-45 b e x^2}{a+b x^4}\right ) \, dx}{384 a^3 b}\\ &=\frac{x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}-\frac{\int \frac{-21 (11 b c+a g)-45 b e x^2}{a+b x^4} \, dx}{384 a^3 b}+\frac{(5 d) \int \frac{x}{a+b x^4} \, dx}{16 a^3}\\ &=\frac{x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{(5 d) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{32 a^3}+\frac{\left (77 b c+15 \sqrt{a} \sqrt{b} e+7 a g\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{256 a^{7/2} b^{3/2}}-\frac{\left (15 e-\frac{7 (11 b c+a g)}{\sqrt{a} \sqrt{b}}\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{256 a^3 b}\\ &=\frac{x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}-\frac{\left (77 b c-15 \sqrt{a} \sqrt{b} e+7 a g\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{512 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\left (77 b c-15 \sqrt{a} \sqrt{b} e+7 a g\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\left (77 b c+15 \sqrt{a} \sqrt{b} e+7 a g\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^{7/2} b^{3/2}}+\frac{\left (77 b c+15 \sqrt{a} \sqrt{b} e+7 a g\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^{7/2} b^{3/2}}\\ &=\frac{x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}-\frac{\left (77 b c-15 \sqrt{a} \sqrt{b} e+7 a g\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\left (77 b c-15 \sqrt{a} \sqrt{b} e+7 a g\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\left (77 b c+15 \sqrt{a} \sqrt{b} e+7 a g\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\left (77 b c+15 \sqrt{a} \sqrt{b} e+7 a g\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}\\ &=\frac{x \left (b c-a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3}+\frac{x \left (7 (11 b c+a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (11 b c+a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}-\frac{\left (77 b c+15 \sqrt{a} \sqrt{b} e+7 a g\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\left (77 b c+15 \sqrt{a} \sqrt{b} e+7 a g\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\left (77 b c-15 \sqrt{a} \sqrt{b} e+7 a g\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\left (77 b c-15 \sqrt{a} \sqrt{b} e+7 a g\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.366398, size = 411, normalized size = 0.94 \[ \frac{-\frac{256 a^{11/4} \sqrt [4]{b} (a (f+g x)-b x (c+x (d+e x)))}{\left (a+b x^4\right )^3}+\frac{32 a^{7/4} \sqrt [4]{b} x (a g+11 b c+b x (10 d+9 e x))}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} \sqrt [4]{b} x (7 a g+77 b c+15 b x (4 d+3 e x))}{a+b x^4}-6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (80 \sqrt [4]{a} b^{3/4} d+15 \sqrt{2} \sqrt{a} \sqrt{b} e+7 \sqrt{2} a g+77 \sqrt{2} b c\right )+6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-80 \sqrt [4]{a} b^{3/4} d+15 \sqrt{2} \sqrt{a} \sqrt{b} e+7 \sqrt{2} a g+77 \sqrt{2} b c\right )-3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )+3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{3072 a^{15/4} b^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 560, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09294, size = 629, normalized size = 1.44 \begin{align*} \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{\sqrt{2}{\left (77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} + \frac{45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} + 7 \, a b^{2} g x^{9} + 126 \, a b^{2} x^{7} e + 160 \, a b^{2} d x^{6} + 198 \, a b^{2} c x^{5} + 18 \, a^{2} b g x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x - 21 \, a^{3} g x - 32 \, a^{3} f}{384 \,{\left (b x^{4} + a\right )}^{3} a^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]